### Player I - Webnode

```•
•
•
•
Name: Trương Hoài Anh
Email: [email protected]
• Class email:
[email protected]
• Pass: qmiu12345678
Chapter 1: Introduction to
Quantitative Analysis
1. Describe the quantitative analysis approach
2. Understand the application of quantitative analysis in
a real situation
3. Describe the use of modeling in quantitative analysis
4. Use computers and spreadsheet models to perform
quantitative analysis
5. Discuss possible problems in using quantitative
analysis
6. Perform a break-even analysis
Chapter 2: Probability

Basic Definitions: Events, Sample Space, and Probabilities
Basic Rules for Probability

Conditional Probability

Independence of Events


Combinatorial Concepts
The Law of Total Probability and Bayes’ Theorem

Random variables

Module 1: Game theory
Two-person means there are two competing
players in the game.
 Zero-sum means the gain (or loss) for one
player is equal to the corresponding loss (or
gain) for the other player.
 The gain and loss balance out so that there is
a zero-sum for the game.
 What one player wins, the other player loses.

Module 1: Game theory
Payoff table
4 steps




Row minimum
Maximin
Column maximum
Minimax
Pure vs Mixed vs Dominated strategies
Best Strategy
For Player II
Example 1
Player II
b1
b2
a1
19
20
19
a2
5
-4
-4
19
20
Player I
Column
maximum
Best Strategy
For Player I
Row
minimum
Maximin
Payoff
Minimax
Payoff
Example 2
Player I holds a black Ace and a red 8. Player II holds
a red 2 and a black 7. The players simultaneously
choose a card to play. If the chosen cards are of the
same color, Player I wins. Player II wins if the cards
are of different colors. The amount won is a number
of dollars equal to the number on the winner’s card
(Ace counts as 1.)
• Establish the payoff table
• Find the value of the game and the optimal
mixed strategies of the players
Player II
Player I
Red 2
Black 7
b1
b2
Black Ace
a1
-2
Red 8
a2
8
1
-7
8
1
Column
maximum
Row
minimum
-2
-7
Expected value
Red 2
b1 (p)
Player I
Black 7
b2 (1-p)
EV
Black Ace
a1 (q)
-2
1
-2p + (1-p)
Red 8
a2 (1-q)
8
-7
8p -7(1-p)
EV
-2q + 8(1-q) q -7(1-q)
• EV for q: -2q+8(1-q)=q-7(1-q) => q=15/18
• EV for p: -2p+(1-p)=8p-7(1-p) => p=4/9
• EV: -2(4/9)+(1-4/9)= -1/3
Dominated strategy
• 2 players, zero-sum
• At least one player has more than 2 options
• Solution
Payoff
4 steps
Pure strategy or mixed strategy
 Pure => EV
 Mixed => elimination => EV
Example 3
X1
X2
X3
Column
maximum
Y1
Y2
13
6
12
0
8
14
13
14
Row
minimum
0
6
12
Example 3
X1
X2
X3
Column
maximum
Y1
Y2
13
6
12
0
8
14
13
14
Row
minimum
0
12
M4.15
• ST Co. and FF Co. are both vying for more share
of the market. If ST does no advertising, it will
not lose any share of market if FF does nothing.
It will lose 2% of market if FF invests \$10,000,
and it will lose 5% if FF invests \$20,000 in
advertising. On the other hand, if ST invests
\$15,000 it will gain 3% if FF does nothing; gain
1% if FF invests \$10,000; and lose 1% if FF
• Develop a payoff table
• Find the value of the game
M4.15 (sol.)
Do
nothing
b1
ST Co.
Do
nothing
\$15,000
\$10,000 \$20,000
b2
b3
Row
minimum
a1
0
-2
-5
-5
a2
3
1
-1
-1
3
1
-1
Column
maximum
```